What Special About This Number
red for the properties in dozenal (dependent in which base is used, in this wiki, we always use the dozenal base). 0 is the additive identity. 1 is the multiplicative identity. 2 is the only even prime. 3 is the only prime which is one less than a square number. 4 is the smallest composite number. 5 is the smallest k such that akxk+ak-1xk-1+...+a1x+a0 does not have algebraic solution. 6 is the smallest number > 1 which is not a prime power. 7 is the smallest k such that regular k-gon is not constructible using compass and straightedge. 8 is the smallest positive k not having primitive roots. 9 is the only positive perfect power that is one more than another positive perfect power. X is the smallest noncototient number. E is the smallest k such that regular k-gon is not constructible using neusis, or an angle trisector. 10 appears in the value of the Riemann zeta function at −1 (i.e. ζ(−1) = −1/10). 11 is the number of Archimedean solids. 12 is the smallest nontotient number. 13 is the smallest k>1 such that k-th cyclotomic polynomial has more terms than the largest prime factor of k. 14 is the only number of the form ab ≠ ba, with a, b nonnegative integers, a != b. 15 is the only positive Genocchi prime. 16 is the smallest base not a perfect power (where Brazilian numbers can be factored algebraically) for which there are no proven Brazilian primes. 17 is the number of hexagons of the only non-trivial normal magic hexagon. 18 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. 19 is the smallest number of distinct squares needed to tile a square. 1X is the numerator of an approximation of π (1X/7). 1E is the smallest number n such that Relative class number h- of cyclotomic field Q(zeta_n) is greater than 1. 20 is the largest number for which the Dirichlet characters are all real. 21 is the smallest number >1 which is both square and centered square. 22 is the only positive number to be directly between a square and a cube. 23 is the number n'' for which (the largest number in the sequence of ''n for Collatz conjecture)/(n''2) is largest. (i.e. 5414/23 = 245.E14) 24 is the only perfect number ''n such that there is no k''≠''n such that σ(k'')−''k = n''. 25 is the largest number ''n such that 2''x''2 + n'' is prime for all 0≤''x≤''n''−1. (since it is divisible by n'' for ''x = n'', one cannot do be better than this) 26 is the largest number with the property that all smaller numbers relatively prime to it are prime or 1. 27 is one of the only two numbers which is a repunit in three or more bases (not including base 1). 28 is the smallest number ''n such that the n''-th row of the modulo-2 Pascal's triangle (the top row, which contains only one 1, is the 0th row, not the 1st row), when read in binary, is not a number of the sides of constructible regular polygon. 29 is the largest number that is not a sum of distinct triangular numbers. 2X is the smallest number with the property that it and its neighbors have the same number of divisors. 2E is the smallest number whose reciprocal does not terminate and has even period length, but does not satisfy Midy's theorem. 30 is the smallest perfect power which is not a prime power. 31 is the smallest irregular prime. 32 is the magic number of the only non-trivial normal magic hexagon. 33 is the smallest ''n ≠ 10 such that n''.''n.n''...''n.n''.1 (dot means concatenation) cannot be prime. 34 is the number of n-queens problem solutions for n = 7. 35 is the largest number ''n such that x''2 + ''x + n'' is prime for all 0≤''x≤''n''−2. (since it is divisible by n'' for ''x = n''−1, one cannot do be better than this) 36 is the largest number of sides of a regular polygon that can fill a point with other regular polygons. 37 is the smallest number ''n such that (define a(n): a(0)=a(1)=1; thereafter a(n+1) = sum(a(k)^2,k=0..n)/n) a(n) is not integer. 38 is the smallest n'' such that all of ''n.0, n''.1, ''n.2, ..., n''.E (dot means concatenation) are composite. (i.e. all of 10''n+0, 10''n''+1, 10''n''+2, ..., 10''n''+E are composite) 39 is the smallest odd positive integer that is not power of squarefree number. 3X is the largest even number which is a value of D for incrementally largest values of minimal x satisfying Pell equation x^2-Dy^2=1. 3E is the smallest number n'' such that ''n×2''k''+1 is composite for all 1≤''k''≤100. 40 is the largest number n'' such that the sum of the first ''n positive triangular numbers is also a triangular number.